Optimal transportation, topology and uniqueness

TL;DR

This paper investigates the conditions under which solutions to the optimal transportation problem are unique, especially when the cost function is smooth and the underlying space has complex topology, by linking it to Birkhoff's extremality problem.

Contribution

It provides a new characterization of extremal measures using numbered limb systems, and applies this to establish solution uniqueness on spherical manifolds.

Findings

Characterization of extremal measures via numbered limb systems.

Necessary and near-sufficient conditions for measure extremality.

Application to uniqueness of solutions on spherical manifolds.

Abstract

The Monge-Kantorovich transportation problem involves optimizing with respect to a given a cost function. Uniqueness is a fundamental open question about which little is known when the cost function is smooth and the landscapes containing the goods to be transported possess (non-trivial) topology. This question turns out to be closely linked to a delicate problem (# 111) of Birkhoff [14]: give a necessary and sufficient condition on the support of a joint probability to guarantee extremality among all measures which share its marginals. Fifty years of progress on Birkhoff's question culminate in Hestir and Williams' necessary condition which is nearly sufficient for extremality; we relax their subtle measurability hypotheses separating necessity from sufficiency slightly, yet demonstrate by example that to be sufficient certainly requires some measurability. Their condition amounts to the vanishing of the measure \gamma outside a countable alternating sequence of graphs and antigraphs in which no two graphs (or two antigraphs) have domains that overlap, and where the domain of each graph / antigraph in the sequence contains the range of the succeeding antigraph (respectively, graph). Such sequences are called numbered limb systems. We then explain how this characterization can be used to resolve the uniqueness of Kantorovich solutions for optimal transportation on a manifold with the topology of the sphere.